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We continue to study quantum complexity classes and various extended models of quantum computing. In this lecture and the next, we examine scenarios under which an algorithm has limited access to an “all-powerful ” computational device, called an oracle, and analyse the effects of this on the algorithm’s computational power. In this lecture, we study the model of computation with advice. To begin, we define what advice means in both classical and quantum computational settings. 1.1 Classical Advice Suppose that we have N input variables x1,...,xN ∈ {0, 1}, and we wish to compute a function f(x1,...,xN) of these variables. In the classical model of computation with advice, the algorithm receives, in addition to the input, some advice string m. The advice string comes from an oracle and can contain information about the correct result of the computation on input x1,...,xN. However, it can depend only on the length N of the input, and not on the specific input itself. (In other words, the advice string must be the same for every input of length N.) We say that a problem f is “solvable in polynomial time with advice of length g(N)” if there exists a polynomial time Turing machine M with the following property: for every input length N, there exists an advice string m of length g(N) such that feeding the Turing machine with the input string x1,...,xN and the advice string m results in the output of the correct answer to the problem on that input. That is: ∃polynomial time M ∀N ∃m ∈ {0, 1} g(N) {M(x1,...,xN, m) = f(x1,...,xN)} If such a Turing machine exists, we write: f ∈ P/g(N) This notation generalizes in the obvious manner by writing other complexity classes in place of P. Typically we will just write poly, instead of a function such as g(N), to denote advice of polynomial size

Year: 2009

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